无穷远分界线环的Melnikov判据及二次系统极限环的分布

THE MELNIKOV CRITERION OF AN INFINITE SEPARATRIX LOOP AND THE DISTRIBUTION OF LIMIT CYCLES IN A QUADRATIC SYSTEM

1963年，Melnikov在[1]中考虑系统x=P_o(x,y)+εP_1(x,y,wt.ε),y=Q_o(x,y)+εQ_1(x,y,wt.ε)

This paper gives an exact condition of applying Melnikovs criterion by defining theconcept of suppressing condition. For the case where the separatrix tends to an infinitesaddle point this paper gives the Melnikov criterion which is similar to the Melnikov criterionin the case of a finite saddle point, and points out the difference between the two cases. Byapplying the new criterion given in this paper the author discusses the bifurcation diagram ofa quadratic system which has a 3-order fine focus. We prove the impossibility of generatinga distribution of limit cycles of the type (0,4) by bifurcation from the fine focus and theinfinite separatrix at the same parameter values. This paper determines the approximatebifurcation value by thoeretical analysis. The bifurcation value thus determined preciselycoincides with the value as determined on a computer.